A random sample of students was surveyed and asked to list their grade level and whether or not they have a pet. Results are shown in the table below, but some values are missing. Fill in the missing values.

Question

Pets Survey
Grade Pets No Pets Total
6th grade 11
23
7th grade
8th grade
10 36
Total
38 91

Answer 1

To fill in the missing values in the table, we need to use the information provided in the totals. Let's represent the missing values with variables:

• Let \( x \) be the number of 7th graders with pets.
• Let \( y \) be the number of 7th graders without pets.
• Let \( z \) be the number of 8th graders with pets.
• Let \( w \) be the number of 8th graders without pets.

From the table, we have the following totals:

• Total number of students with pets: 38
• Total number of students without pets: 91
• Total students in 6th grade: 23
• Total students in 8th grade: 36

Let's analyze the table and fill in the gaps step by step:

1. 6th grade pets: There's already 11 students with pets.

   • Total 6th graders: 23
   • Therefore, No Pets in 6th grade: \( 23 - 11 = 12 \)

2. Total pets: This is given as 38.

   • We have 11 from 6th grade, plus \( x \) from 7th grade, plus \( z \) from 8th grade.
   • So, we have:
     \[ 11 + x + z = 38 \]

   Rearranging gives:
   \[ x + z = 27 \quad (1) \]

3. Total No Pets: This is given as 91.

   • We have 12 from 6th grade, plus \( y \) from 7th grade, plus \( w \) from 8th grade.
   • So, we have:
     \[ 12 + y + w = 91 \]

   Rearranging gives:
   \[ y + w = 79 \quad (2) \]

4. Total for 8th graders: This is given as 36.

   • Therefore, we can say:
     \[ z + w = 36 \quad (3) \]

Now we have a system of equations (1), (2), and (3). From equations (1) and (3), we can express \( z \) and \( w \) in terms of \( x \) and \( y \):

From equation (3):
\[ w = 36 - z \quad (4) \]

Replace \( z \) in (1):
\[ x + (36 - w) = 27 \quad \Rightarrow \quad x + 36 - w = 27 \quad \Rightarrow \quad x - w = -9 \quad \Rightarrow \quad x = w - 9 \quad (5) \]

Now insert (5) into (2):
\[ (w - 9) + w = 79 \quad \Rightarrow \quad 2w - 9 = 79 \quad \Rightarrow \quad 2w = 88 \quad \Rightarrow \quad w = 44 \]

Now, substituting \( w = 44 \) back to find \( z \):
Using equation (3):
\[ z + 44 = 36 \quad \Rightarrow \quad z = 36 - 44 = -8 \quad (This is incorrect. Let's re-check) \]

Substituting correctly into \( y + w = 79 \):
So \( y = 79 - 44 = 35 \).

Now substituting \( 44 \) into \( w = 36 - z \) we find:
\[ z + 44 = 36 \implies z = 36 - 44 = -8 ; (impossible) \]

Let’s correct and find correct counting. We need better relations.

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Let me know your thoughts!